• ### Unit 7: Work and Energy

Energy describes the capacity of a physical system to perform work. It plays an essential role in everyday events and scientific phenomena. You can probably name many forms of energy: from the energy our food provides us, to the energy that runs our cars, to the sunlight that warms us on the beach. Not only does energy have many interesting forms, but it is involved in almost all phenomena and is one of the most important concepts of physics.

Energy can change forms, but it cannot appear from nothing or disappear without a trace. Thus, energy is one of a handful of physical quantities that we say is conserved.

Completing this unit should take you approximately 5 hours.

• ### 7.1: Calculating Work and Force

Work is done on a system when a constant applied force causes the system to be displaced or moved in the direction of the applied force. We can describe work using the equation $W = Fd\cos\theta$, where $F$ is force, $d$ is displacement, and $\theta$ (the Greek letter theta) is the angle between $F$ and $d$.

From the equation for work, we can see that the unit for work must be the Newton-meter: the unit for force is the Newton and the unit for displacement (distance) is the meter. We define the Newton-meter as the unit joule. Consequently, we use joules as the unit for work and energy.

• ### 7.2: Work, Potential Energy, and Linear Kinetic Energy

We define kinetic energy as the energy associated with motion. We calculate kinetic energy as ${KE} = \frac{1}{2}mv^2$.

When work is done on a system, energy is transferred to the system. We define net work as the total of all work done on a system by all external forces. We can think of the sum of all the external forces acting on a system as a net force, or $F_{\mathrm{net}}$.

We can write the equation for net work in a similar way to how we wrote the equation for work earlier: $W_{\mathrm{net}} = F_{\mathrm{net}}d\cos\theta$, where $W_{\mathrm{net}}$ is net work, $F_{\mathrm{net}}$ is net force, $d$ is displacement, and $\theta$ is the angle between force and displacement.

• ### 7.3: Conservative Forces and Potential Energy

A non-conservative force is a force that depends on the path an object takes. In other words, a non-conservative force depends on how an object got from its initial state to its final state. Non-conservative forces change the amount of mechanical energy in a system. This differs from conservative forces, which do not depend on the path taken from initial to final state, and do not change the amount of mechanical energy in a system.

An important example of a non-conservative force is friction. We know that friction is the force between two surfaces. We see friction when rolling a ball on a carpet versus a hardwood floor. The ball rolls farther on the hardwood floor than it does on a carpet. This is because the fuzzy carpet has more friction than the smooth hardwood. Friction converts some of the kinetic energy of the ball to thermal energy, or heat. As kinetic energy is converted to thermal energy, the balls slows to a stop.

On the other hand, a conservative force is a force which does work that only depends on the beginning point and the end point of the system. The work done by a conservative force does not depend on the path the system takes to get from beginning to end. Conservative forces exist in ideal systems with no friction. An idealized spring that does not experience friction would be an example of conservative forces.

• ### 7.4: Conservation of Energy

The Law of Conservation of Energy states that the total energy in any process is constant. Energy can be transformed between different forms, and energy can be transferred between objects. However, energy cannot be created or destroyed. This is a broader law than the conservation of mechanical energy because this applies to all energy, not just energy when only conservative forces are applied.

We can write the Law of Conservation of Energy as $KE + PE + OE = \mathrm{Constant}$ or as $KE_{i} + PE_{i} + OE_{i} = KE_{f} + PE_{f} + OE_{f}$

In the second equation, the $KE_{i}$, $PE_{i}$, and $OE_{i}$ are initial conditions and $KE_{f}$, $PE_{f}$, and $OE_{f}$ are final conditions. The new term, $OE$, is other energy. This is a collected term for all forms of energy that are not kinetic energy or potential energy. Other forms of energy include: thermal energy (heat), nuclear energy (used in nuclear power plants), electrical energy (used to power electronics), radiant energy (light), and chemical energy (energy from chemical reactions).

When solving Conservation of Energy problems, it is important to identify the system of interest, and all forms of energy that can occur in the system. To do this, we need to first identify all forces acting on the system. Then, we can plug equations for different types of energy into the Law of Conservation of Energy equation to solve for the unknown in the problem.

• ### 7.5: Rotational Kinetic Energy

Why do tornadoes spin so rapidly? The answer is that the air masses that produce tornadoes are themselves rotating, and when the radii of the air masses decrease, their rate of rotation increases. An ice skater increases their spin in an exactly analogous way. The skater starts their rotation with outstretched limbs and increases their spin by pulling them in toward their body. The same physics describes the spin of a skater and the wrenching force of a tornado. Clearly, force, energy, and power are associated with rotational motion.

We cover these and other aspects of rotational motion in this unit. We will see that important aspects of rotational motion have already been defined for linear motion or have exact analogs in linear motion.

We can write an equation for the rotational kinetic energy (the energy of rotational motion) as: $KE_{\mathrm{rot}}=\frac{1}{2}I \omega^{2}$

• ### 7.6: Power

We define power as the rate at which work is done. We can write this as $P = \frac{W}{\Delta t}$, where $w$ is work and $\Delta t$ is the duration of the work being done. The unit for power is the watt, W. One watt equals one joule per second.

Higher power means more work is done in a shorter time. This also means that more energy is given off in a shorter time. For example, a 60 W light bulb uses 60 J of work in a second, and also gives off 60 J of radiant and heat energy every second.